The bowl, ring, and cage structures of C-20 have been studied using the B3LYP (Becke 3-parameter-Lee-Yang-Parr) method. Fingerprint-quality infrared spectra have been computed for all species. The computed geometries differ drastically from those obtained using the local density approximation (LDA). Combining coupled cluster results at LDA and SCF geometries with the present B3LYP results, and including zero-point energies, we find the bowl structure to be the lowest in energy, two distinct cage structures with C-i and C-2h symmetry both about 11 kcal/mol higher, and a C-10h ring about 47 kcal/mol higher. Statistical thermodynamical calculations reveal that the bowl will dominate up to about 1200 K, and the ring above 1900 K. Despite being much lower in energy than the ring, the cage structure is not present in appreciable proportion due to the extra vibrational entropy the ring derives from many extremely low-lying vibrations.